Decidability and Complexity Results for Timed Automata and Semi-linear Hybrid Automata

  • Joseph S. Miller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1790)

Abstract

We define a new class of hybrid automata for which reach-ability is decidable—a proper superclass of the initialized rectangular hybrid automata—by taking parallel compositions of simple components. Attempting to generalize, we encounter timed automata with algebraic constants. We show that reachability is undecidable for these algebraic timed automata by simulating two-counter Minsky machines. Modifying the construction to apply to parametric timed automata, we reprove the undecidability of the emptiness problem, and then distinguish the dense and discrete-time cases with a new result. The algorithmic complexity—both classical and parametric—of one-clock parametric timed automata is also examined. We finish with a table of computability-theoretic complexity results, including that the existence of a Zeno run is Σ11-complete for semi-linear hybrid automata; it is too complex to be expressed in first-order arithmetic.

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References

  1. 1.
    Rajeev Alur and David L. Dill. A theory of timed automata. Theoret. Comput. Sci., 126(2):183–235, 1994.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Rajeev Alur, Thomas A. Henzinger, and Moshe Y. Vardi. Parametric real-time reasoning. In Proceedings of the Twenty-Fifth Annual ACM Symposium on the Theory of Computing, pages 592–601, San Diego, California, 16–18 May 1993.Google Scholar
  3. 3.
    Eugene Asarin and Oded Maler. Achilles and the tortoise climbing up the arithmetical hierarchy. Journal of Computer and System Sciences, 57(3):389–398, December 1998.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Olivier Bournez. Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy. Theoretical Computer Science, 210(1):21–71, January 1999.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. G. Downey and M. R. Fellows. Parameterized complexity. Springer-Verlag, New York, 1999.Google Scholar
  6. 6.
    O. A. Gross. Preferential arrangements. Amer. Math. Monthly, 69:4–8, 1962.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Thomas A. Henzinger, Peter W. Kopke, Anuj Puri, and Pravin Varaiya. What’s decidable about hybrid automata? J. Comput. System Sci., 57(1):94–124, 1998. 27th Annual ACM Symposium on the Theory of Computing (STOC’95) (Las Vegas, NV).MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Y. Kesten, A. Pnueli, J. Sifakis, and S. Yovine. Integration graphs: a class of decidable hybrid systems. In Hybrid systems, pages 179–208. Springer, Berlin, 1993.Google Scholar
  9. 9.
    M. Kourjanski and P. Varaiya. A class of rectangular hybrid systems with computable reach set. Lecture Notes in Computer Science, 1273:228–234, 1997.CrossRefGoogle Scholar
  10. 10.
    G. Lafferriere, G. J. Pappas, and S. Sastry. O-minimal hybrid systems. Technical Report UCB/ERL M98/29, Department of Electrical Engineering and Computer Science, University of California at Berkeley, May 1998.Google Scholar
  11. 11.
    Marvin L. Minsky. Recursive unsolvability of Post’s problem of “tag” and other topics in theory of Turing machines. Ann. of Math. (2), 74:437–455, 1961.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Anuj Puri and Pravin Varaiya. Decidability of hybrid systems with rectangular differential inclusions. In Computer aided verification (Stanford, CA, 1994), pages 95–104. Springer, Berlin, 1994.Google Scholar
  13. 13.
    Alfred Tarski. A Decision Method for Elementary Algebra and Geometry. RAND Corporation, Santa Monica, Calif., 1948.MATHGoogle Scholar
  14. 14.
    Lou van den Dries. Tame topology and o-minimal structures. Cambridge University Press, Cambridge, 1998.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Joseph S. Miller
    • 1
  1. 1.Department of MathematicsCornell UniversityIthaca

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